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Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes

Abstract : In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Amp\`ere measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-P\u{a}un \cite{DP04} and Boucksom-Demailly-P\u{a}un-Peternell \cite{BDPP13}. Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The results obtained here will be used in \cite{GL21b} for solving degenerate complex Monge-Amp\`ere equations on compact Hermitian varieties.
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https://hal-universite-paris-saclay.archives-ouvertes.fr/hal-03288253
Contributor : Hoang-Chinh Lu <>
Submitted on : Wednesday, July 28, 2021 - 2:02:17 PM
Last modification on : Thursday, July 29, 2021 - 10:43:56 AM

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  • HAL Id : hal-03288253, version 1
  • ARXIV : 2106.04272

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Vincent Guedj, Chinh H. Lu. Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes. 2021. ⟨hal-03288253⟩

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